Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces

2025/07/04 by Kannaka, Kazuki, Kobayashi, Toshiyuki
#22E40 #22E46 #53C30 #58H15. Secondary:22D50 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Primary:57S30

paper · doi:10.48550/arxiv.2507.03476

Abstract

Let X=G/H be a homogeneous space of a Lie group G. When the isotropy subgroup H is non-compact, a discrete subgroup Γ may fail to act properly discontinuously on X. In this article, we address the following question: in the setting where G and H are reductive Lie groups and Γ\backslash X is a standard quotient, to what extent can one deform the discrete subgroup Γ while preserving the proper discontinuity of the action on X? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients Γ\backslash X, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur.

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